1 Introduction

This document estimates yearly-trends in the Proportion of Illegally Killed Elephants (PIKE) from MIKE (Monitoring Illegally Killed Elephants) monitoring sites in Africa since 2003. The method used here was published in 2020 on a GitHub repository CITESmike2020/MIKE-GLMM with the computer code (written in R) and several accompanying technical reports. The original technical report, which this document is based on, can be viewed by clicking here.

The computer code from the GIT hub repository has been modified to estimate yearly PIKE trends from 2003-2021 using only the unweighted marginal mean PIKE model (MM.p.uw), which does not require elephant population abundance data at the site-year level. The pro and cons of this approach are discussed in more detail in the original technical document.

Briefly, MIKE data is collected on an annual basis in designated MIKE sites by law enforcement and ranger patrols in the field and through other means. When an elephant carcass is found, site personnel try to establish the cause of death and other details, such as sex and age of the animal, status of ivory, and stage of decomposition of the carcass. This information is recorded in standardized carcass forms, details of which are then submitted to the MIKE Programme. As expected, different sites report widely different numbers of carcasses, as encountered carcass numbers are a function of: population abundance; natural mortality rates; the detection probabilities of elephant carcasses in different habitats; differential carcass decay rates; levels of illegal killing; and levels of search effort and site coverage. Because of these features of the survey data, the number of carcasses found is unlikely to be proportional to the total mortality and trends in observed numbers of illegally killed elephants may not be informative of the underlying trends. Consequently, the observed proportion of illegally killed elephants (PIKE) as an index of poaching levels has been used in the MIKE analysis in an attempt to account for differences in patrol effort between sites and over time: \[PIKE_{sy}=\frac{\textit{Number of illegally killed elephants}_{sy}}{\textit{Total Carcasses Examined}_{sy}}\] where the subscripts \(sy\) refer to site and year respectively.

Computing a continent-wide PIKE is challenging for several reasons, including as mentioned above:

2 Exploration of PIKE data

2.1 MIKE sites with PIKE data

There are 69 MIKE sites that have reported data on the number of carcasses found and the number of illegally killed carcasses among these. This includes 64 site-years where sites have reported 0 carcasses and 805 site-years where sites have reported one or more elephant carcasses.

The current analysis treats a site that did not detect and report any carcasses in a year due to no patrol effort and a site that did conduct patrols but did not detect any carcasses in a year in the same manner. This is because information on patrol effort is not currently used in the analysis and only the number of carcasses detected and examined, and the number of illegally killed elephants in the sample of carcasses is used. In the latter case, 0 illegal carcasses out of 0 carcasses examined gives a PIKE for that site-year of 0/0 which is indeterminate and cannot be used in any mathematical analysis of PIKE. The following plot shows that there are some sites that have reported data for at least one carcass in as little as one year, but other sites have reported data for at least one carcass in almost every year.

In total, there are 805 unique site-years in Africa since 2003 where data has been reported (and the number of reported carcasses > 0).

The number of carcasses reported in each site-year since 2003 varies enormously from 1 to 351 carcasses.

The unusual data point for West Africa where one site reported a large number of carcasses in one year is correct and corresponds to the MIKE site Gourma (GOU) with total number of carcasses equal to 134, of which 130 were poached by armed groups who entered the area.

2.2 Observed PIKE

The observed PIKE is the value computed from the examined carcasses in a year which we hope reflects the actual PIKE for all elephants at the MIKE site. A plot of the observed PIKE values from each site-year shows a wide range in the observed PIKE values, but many of the observed PIKE values close to 0 or 1 occur in sites with only a small number of carcasses examined in a year:

The trend in observed PIKE values for each site is:

Note that with a small number of carcasses reported (e.g. 0 or 1) it is quite common for the reported PIKE to be 0 or 1 because either none or all of the carcasses have been illegally killed. Consequently, the trends are difficult to interpret for many sites with only a few carcasses reported.

2.3 Final dataset used

The final data set consists of 64 MIKE sites from 2003 to 2021 over the subregions as shown below:

Summary of MIKE sites used in analysis
Subregion Name Number of sites # Site-Years Mean # carcasses reported per year Site IDs
Central Africa 14 206 19.8 BBK, BGS, DZA, GAR, LOP, MKB, NDK, ODZ, OKP, SAL, SGB, VIR, WAZ, ZAK
Eastern Africa 15 234 41.2 AKG, BBL, EGK, GSH, KSH, KTV, MCH, MKZ, MRU, QEZ, RHR, SBR, SEL, TGR, TSV
Southern Africa 18 203 43.0 CHE, CHO, ETO, HWA, KFE, KRU, KSG, LPP, LZN, MAG, MAN, MJT, NIA, NLW, NYA, SLW, SMN, ZBZ
West Africa 17 162 5.8 COM, FAZ, GOU, KAK, KER, MAR, MOL, NAZ, NKK, PDJ, SAP, TAI, WBF, WBJ, WNE, YKR, ZIA

3 The Bayesian model

3.1 Binomial variation within each site-year

In each site-year, the number of illegally killed elephant carcasses is a fraction of the total elephant carcasses examined. Consequently, we use a binomial distribution to model this part of the data:

\[IC_{sy} \sim Binomial(TC_{sy}, \pi_{sy})\] where \(IC_{sy}\) is the number of illegally killed carcasses reported from site \(s\) in year \(y\); \(TC_{sy}\) is the total number of carcasses located as reported from site \(s\) in year \(y\); and \(\pi_{sy}\) is the probability that a reported carcass was defined as illegally killed in site \(s\) in year \(y\).

3.2 Temporal and site effects

The value of \(\pi_{sy}\) (the PIKE in site \(s\) and year \(y\)) varies by time (temporal trends), by site (site effects) and over time within each site (site-year effects).

Because, the PIKE must be between 0 and 1, it is modelled on the logistic scale. Similar to (but not exactly the same as) Burn, Underwood and Blanc (2011), a Bayesian hierarchical model is adopted of the form: \[logit(\pi_{sy})= Year_y + Site_s(R) + SiteYear_{sy}(R)\] where \(Year_y\) is the effect of year \(y\) on the \(logit(PIKE)\); \(Site_s(R)\) is the (random) effect of site \(s\) on the \(logit(PIKE)\); and \(SiteYear_{sy}(R)\) is the (random) effect of site \(s\) in year \(y\) on the \(logit(PIKE)\).

Here \(year\) is not modelled in a hierarchical fashion because we are interested in these particular years and do not believe that these years represent a (theoretical) sample from all possible years.

The random effects of site and site-year are modelled using a hierarchical model, i.e. \[Site_s \sim Normal(0, \sigma_{site})\] and \[SiteYear_{sy} \sim Normal(0, \sigma_{site.year})\]

Here the \(Year_y\) effects represents the average \(logit(PIKE)\) over all sites giving each site an equal weight.

3.3 Marginal mean PIKE

Once the model is fit, the estimated \(logit(PIKE)\) for all sites and years where no data are collected is found as: \[\widehat{logit(\pi_{sy})}= \widehat{Year}_y + \widehat{Site}_s + \widehat{SiteYear}_{sy}\] Note that if no data are collected in a particular site-year, the estimated PIKE is based purely on the estimated value from other years. Because all \(Site.Year\) effects are assumed to be independent among and within sites, so their values must be simulated from the posterior distribution.

Once the estimated site-year values are obtained, the marginal means are found in two ways:

  1. The marginal mean on the logit scale \[MM_y^{logit} = \frac{\sum_s {\widehat{logit(\pi_{sy})} }}{s}\] where \(s\) is the number of sites.

This marginal mean can also be interpreted as the \(logit(PIKE)\) when the \(Site\) and \(Site.Year\) effects are zero, i.e. for an ``average site’’.

This marginal mean can be back transformed to the [0,1] scale. Because the \(logit()\) scale is a non-linear transformation of the [0,1] scale, this (default) method of computing a marginal mean is greatly influenced by \(logit()\) values from PIKE that are close to 0 or 1, i.e., \(logit(0)=-\infty\) and \(logit(1)=+\infty\). Consequently, this marginal mean is not recommended for use.

  1. The marginal mean on the probability (i.e. the 0-1) scale \[MM_y^{unweighted} = \frac{\sum_s{\widehat{\pi}_{sy}}}{s}\] This is closest to the marginal means computed in the prior analysis (the LSMeans approach) and is the recommended approach for computing the unweighted marginal mean.

3.4 Uncertainty about marginal mean PIKE

There are three sources of uncertainty that need to be considered when estimating the uncertainty about the marginal mean PIKE:

  • choice of MIKE sites
  • imputation of missing PIKE in year.sites where no data is collected
  • estimation of PIKE in a year.site when only a small number of carcasses is measured.

If you believe that MIKE sites were chosen at random from a larger population of MIKE sites and you need to account for this initial selection of sites, then all three sources of uncertainty need to be incorporated into the estimates.

However, MIKE sites were selected to be representative of most major populations of elephants and the notion of a new sample of MIKE sites may not be realistic. In this case, the MIKE sites are ``fixed’’ and only the last two elements of uncertainty need to be incorporated.

The differences between these two interpretations can be made clearer if asked what uncertainty should be reported if all MIKES reported in all years and had perfect information, i.e. the mortality of every single mortality in the associated population is known. If you believe that the current MIKE sites are a random sample from many potential MIKE sites, then there is sampling uncertainty associated with the marginal mean. If you believe that the current set of MIKE sites is fixed and representative, then marginal mean PIKE would then have an uncertainty of 0.

This issue is explored in more detail in Appendix 2 in the original technical document.

It turns out that finding the uncertainty when MIKE sites are treated as “fixed” is automatically provided by the Bayesian analysis and no further computations are needed.

If the MIKE sites are to be treated as a random sample of sites taken from a larger population of MIKE sites, then the Bayesian uncertainty associated with the Year.eff term on the logit scale automatically incorporates all three sources of uncertainty. However, as noted previously and later in the document, you cannot simple take the anti-logit of the Year.eff to get the marginal mean PIKE on the [0,1] scale with the proper accounting of uncertainty because of the transformation bias induced by the anti-logit transform.

We derived the uncertainty of the marginal mean PIKE on the [0,1] scale accounting for a random sample of sites and correcting for the transformation bias, by using Bayesian Bootstrapping (Rubin, 1981; https://stats.stackexchange.com/questions/181350/bootstrapping-vs-bayesian-bootstrapping-conceptually). For each sample from the posterior, the year.site values for PIKE on the logit scale (accounting for uncertainty from a sample of carcasses and imputation for missing year.site values), are converted to the [0,1] scale. A sample of weights is generated from a Dirichlet distribution with prior weights all set to 1. The sample of weights are then used to compute a weighted average of the year.site values on the [0,1] scale.

More formally, \[\textbf{w}\sim Dirichlet(1,1,1,....1_{Nsites})\] \[MM_y^{BB,unweighted} = \sum_s{w_i \times \widehat{\pi}_{sy}}\] The posterior distribution of the Bayesian bootstrap estimator will then account for all sources of uncertainty.

6 Model assessment

We performed model assessments of the model at the continental level and expect that similar findings will occur at the sub-regional levels.

6.1 Mixing of chains

The Gelman and Rubin’s potential scale reduction factor statistic (\(\widehat{R}\); Gelman et al, 2013) measures the relative variation in an estimated parameter among the multiple chains and the variation within a chain. Models should have value of \(\widehat{R}\) close to 1 indicating that the posterior space covered by each chain is very similar. The effective sample size is an adjustment to the number of samples in the posterior for autocorrelation. If successive samples from the posterior have a high autocorrelation, then 10 samples from the posterior provide only incremental information over a single sample from the posterior. The effective sample should be reasonably large for all posterior samples to ensure that the posterior mean, standard deviation, and credible intervals are well estimated.

We examined \(\widehat{R}\) and the effective sample size for several parameter sets:

Rhat and Effective sample size for several parameter sets
Effect Max Rhat Max N.eff Min N.eff
SD Site effects 1.010 220 220
SD Year Site effects 1.003 1100 1100
Site Effects 1.121 4500 23
Year Effects 1.005 4500 420

Mixing appears to be adequate with small values of \(\widehat{R}\) in all parameter sets.

The effective sample size is small (<500) for 2 sites. The sites with small effective sample sizes are:

Sites with small effective sample sizes
MIKE site Avg PIKE Site effect Rhat N eff
ETO 0.01 -5.29 1.121 23
HWA 0.01 -2.94 1.006 380

Sites with small effective sample sizes, tend to have PIKE that are very much larger or very much smaller than the average PIKE as estimated by their site effect. In particular, a site with a PIKE close to 0 or 1 will have a site effect with very small uncertainty and so repeated samples from the posterior will all be very similar. Mixing was adequate (as measured by \(\widehat{R}\)) and so these low effective sample sizes are acceptable.

6.2 Examination of trace plots

Trace plots were constructed for the yearly estimates of PIKE on the logit scale:

Similarly, trace plots were constructed for the estimated standard deviation of the \(site\) and \(site.year\) effects on the logit scale:

All plots show good evidence of mixing of the three chains sampled from the posterior.

6.3 Omnibus goodness of fit

An omnibus goodness-of-fit test can be constructed using Bayesian Predictive Plot (Gellman et al, 2013). For each sample from the posterior, the Tukey-Freeman statistic (Freeman and Tukey, 1950) is computed using the observed data and a simulated data based on the posterior sample. The Tukey-Freeman statistic is less sensitive to small observed and expected values than the usual chi-square goodness-of-fit test.

For example, for a particular value of the posterior sample, the observed Tukey-Freeman statistic is found as the difference between the observed number of illegally killed elephants and the expected number of illegally killed elephants: \[TF.obs = \sum_{site.years}{ (\sqrt{IC_{site.year}}-\sqrt{TC_{site.year}\times\pi_{site.year}})^2}\] The simulated Tukey-Freeman statistic is found as the difference between a simulated number of illegally killed elephants and the expected number of illegally killed elephants: \[IC.sim_{site.year} \sim Binomial( TC_{site.year}\times \pi_{site.year})\] \[TF.sim = \sum_{site.years}{ (\sqrt{IC.sim_{site.year}}-\sqrt{TC_{site.year}\times \pi_{site.year}})^2}\] The value of the \(TF.obs\) is plotted against the corresponding \(TF.sim\) and the proportion of times that the observer Tukey-Freeman statistic exceeds the simulated Tukey-Freeman statistic is known as the Bayesian p-value. If the model fits well, then these two measures should be similar and the Bayesian p-value will be close to 0.5. If there is lack of fit, then the two measures will be discordant, and the Bayesian p-value will be close to 0 or 1.

The Bayesian Posterior Predictive plot for the omnibus goodness of fit is:

Because the Bayesian p-value is not extreme, the fit is deemed acceptable.

6.4 Over dispersion

6.4.1 General over dispersion

A general measure of over dispersion is to compute a statistic that compares the expected number of illegally killed elephants based on the fitted site-year PIKE with the observed number of illegally killed elephants.

\[Disperson = \sum_{sy}{\frac{(TC_{sy}\times\widehat{\pi}_{sy}-IC_{sy})^2}{TC_{sy}\times\widehat{\pi}_{sy}}}\] There are 805 site-year data points in the sum above.

This is traditionally divided by the \((\textit{number of data points} - \textit{the number of estimated parameters})\). However, in Bayesian hierarchical models (such as this), the number of parameters is ill-defined. For example, we model site-effects as random variables from a common distribution. Is the number of parameters 2 (the mean and variance of the common distribution) or is it the number of sites (we need to estimate the individual site effects). Furthermore shrinkage in Bayesian models implies that the effective number of site estimates is smaller than the number of sites. A similar problem occurs with the site-year effects. If you count the individual year effects, the individual site effects, and the individual site-year effects as separate parameters, this gives a total parameter count of 888 which is more than the number of data points.

The Bayesian output includes a measure \(pD\) defined as the effective number of parameters, i.e. after accounting for shrinkage. We obtain \(pD\)=782.3 which is considerably less and accounts for shrinkage (Spiegelhalter et al. 2002). This gives an over dispersion value of \[OD = \frac{Dispersion}{\textit{\# data points}-\textit{pD}}\] which gives \(OD=\) 5.7. This value is slightly above 2 indicating some evidence of over dispersion, but generally speaking is acceptable.

Some of the expected number of illegally killed elephants are very small which can inflate the numerator. A histogram of the individual components of the Dispersion numerator:

shows that the fit is generally good, with only a few site years where the contribution is large. The (few) site-years where the observed dispersion component is > 1 are shown below and are acceptable in terms of goodness of fit.

Site-years with largest discrepancy in fit
Site ID Year Total number of carcasses Number of Illegal Carcasses Observed PIKE Estimated PIKE Estimated Number of Illegal Carcasses Contribution to dispersion
GAR 2020 3 0 0.00 0.34 1.01 1.01
SAL 2003 2 0 0.00 0.51 1.02 1.02
QEZ 2008 6 0 0.00 0.17 1.02 1.02
YKR 2015 2 0 0.00 0.52 1.03 1.03
NYA 2021 17 0 0.00 0.06 1.03 1.03
PDJ 2010 6 0 0.00 0.17 1.05 1.05
ZIA 2012 2 0 0.00 0.54 1.07 1.07
VIR 2004 33 1 0.03 0.08 2.73 1.10
ETO 2003 21 1 0.05 0.02 0.36 1.11
NYA 2018 10 0 0.00 0.12 1.15 1.15
BBK 2006 12 0 0.00 0.10 1.19 1.19
WBF 2019 4 0 0.00 0.30 1.20 1.20
PDJ 2019 18 0 0.00 0.07 1.27 1.27
SMN 2018 8 0 0.00 0.16 1.29 1.29
MAG 2009 5 0 0.00 0.28 1.38 1.38
ZIA 2010 6 0 0.00 0.24 1.42 1.42
NIA 2020 11 0 0.00 0.13 1.45 1.45
CHO 2016 121 0 0.00 0.01 1.63 1.63
NDK 2013 10 0 0.00 0.17 1.69 1.69
GOU 2010 27 0 0.00 0.07 1.87 1.87
TAI 2004 2 2 1.00 0.39 0.78 1.89
ODZ 2005 73 0 0.00 0.03 2.05 2.05

These generally occur when no illegally killed elephants are reported with an intermediate number of total carcasses reported where the model predicts a non-zero PIKE. Refer to the earlier sections to look at the individual sites reported here.

6.4.2 Overdispersion due to 0 counts

The omnibus test is a general goodness-of-fit measure. The same logic can be used to investigate specific aspects of the fit. In particular, the number of times that the number of illegally killed elephants is reported as 0 is examined.

There were 147 cases over all sites and all years where the number of illegally killed elephant carcasses was reported as zero. After fitting the model, for each sample from the posterior, we simulate the number of illegally killed elephants in the same way as in the omnibus goodness of fit: \[IC.sim_{site.year} \sim Binomial( TC_{site.year}\times\pi_{site.year})\] and count the number of times a count of 0 is obtained. This is compared to the observed number of times a 0 is obtained.

The number of 0 counts is on the higher side, but not unusual relative to that seen from simulated data.

6.5 Spatial correlation among site effects

The (random) site effects have been modelled as independent random effects without explicitly accounting for the spatial structure of the data. However, we find that sites that are close geographically have similar site effects.

Sites that have PIKE consistently above the continental average are labelled as Above the mean; sites that have PIKE consistently below the continental average are labelled as Below the mean.

We notice that sites that are close geographically tend to have similar site effects (size of dot) and in the same direction (above or below the mean, color of dots). This implies there is a spatial correlation among the site effects that has not been directly accounted for in the analysis.

The current analysis is still valid, but inefficient because it has not used the spatial correlation to improve inference. If spatial autocorrelation is explicitly modelled, then information is shared among sites that are geographically close, i.e., if the PIKE increases in one site, then spatial autocorrelation would imply that it would tend to also increase in a nearby site. Of course, if the sites are in different countries with different levels of enforcement or other covariates that impact PIKE, an explicit spatial autocorrelation could introduce a spurious relationship between the PIKE in the two sites unless these other factors (law enforcement etc.) are also modelled. The explicit spatial autocorrelation models rapidly become more complex to account for these features.

Because the current analysis treats all sites as independent (rather than spatially correlated), the uncertainty in the overall yearly PIKE is slightly smaller than from a model with explicity spatial autocorrelation because the effective number of sites used in computing the overall yearly PIKE is smaller when autocorrelation is explicitly modelled. This in turn, implies that the uncertainty of a trend (e.g. trend in the last five years) in the currently model may be slightly understated as well and the posterior belief in a trend will be higher in the current model compared to the model with an explicity spatial autocorrelation. We believe such effects are minor given the spase data at many sites, the large amount of missing site.years and the potential breaking of spatial autocorrelation across country borders.

A potential improvement to the current analysis may be to add another level of random effects (country effects) so that points from the same country that have related site effects then experience a common country effect. This model is currently under investigation.

6.6 Site effects vs number of carcasses observed

A plot of the estimated site effects vs. the total number of carcasses observed over the year is:

This plot shows that the uncertainty in the site effects declines with the total number of carcasses observed (as expected), and a random scatter about 0 (also as expected). There are a few MIKE sites with extreme site effects as labelled in the plot.

6.7 Autocorrelation in year.site effects

This model assumes that \(Year.Site\) effects are independent from year-to-year. However, local effects may last for several years, and so there may be autocorrelation present in the \(Year.Site\) effects.

A plot of the \(Year.Site_i\) vs. the \(Year.Site_{i-1}\) (i.e. a lag 1 plot) is:

shows a very model correlation over time which is sufficiently small that is not a problem. Note that only those site-years where data are collected are used in the above plot.

6.8 Year.Site effect for each site

A plot of the \(Year.Site\) effect for each site:

shows that only a few years had PIKE values within a site that could be considered unusual for that site.

6.9 Observed vs. predicted PIKE

A plot of observed PIKE in each year.site vs. the predicted PIKE is:

The fit is generally very good. For site-years where the number of carcasses was very small (\(<10\)) and the observed PIKE was 0 or 1, the estimated PIKE is pulled towards the yearly average for that year. For site-years with large number of carcasses (\(>25\)) the estimated PIKE matches closely with the observed PIKE. For site-years with intermediate number of carcasses, the estimates are shrunk slightly towards the mean for that year.

This can also be seen in the plots of observed and fitted PIKE for the individual MIKE sites:

There are several interesting patterns that illustrate the features of the model. In years with many carcasses reported, the estimated site-year PIKE will closely match the observed site-year PIKE. In years with few carcasses reported, the estimated site-year PIKE will be pulled towards the continental trend after accounting for the observed relationship between this sites PIKE and the continental trend.

7 References

Burn, R.W., Underwood, F.M., Blanc J. (2011). Global Trends and Factors Associated with the Illegal Killing of Elephants: A Hierarchical Bayesian Analysis of Carcass Encounter Data. PLoS ONE 6(9): e24165. https://doi.org/10.1371/journal.pone.0024165

Chen, Ming-Hui, and Qi-Man Shao. (1999). Monte Carlo Estimation of Bayesian Credible and HPD Intervals. Journal of Computational and Graphical Statistics 8, 69-92. doi:10.2307/1390921.

Freeman, M.F. & Tukey, J.W. (1950). Transformations related to the angular and square root. Annals of Mathematical Statistics, 221, 607–611.

Gelman, A, Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A. and Rubin, D.R. (2013). Bayesian Data Analysis, 3rd Edition. Chapman and Hall/CRC.

Gimenez, O., Morgan, B.J., and Brooks, S. (2009). Weak identifiability in models for mark-recapture-recovery data. pp.1055-1068 in Thomson, Cooch and Conroy (eds) Modeling demographic processes in marked populations. Springer.

lsmeans (2019). pike TREND ANALYSIS USING THE LEAST-SQUARES MEANS APPROACH in R. https://github.com/CITES-MIKE/MIKE-LSMEANS

Lunn, D., Jackson, C., Best, N., Thomas, A. and Spiegelhalter, D. (2012). The BUGS Book – A practical introduction to Bayesian Analysis. Chapman and Hall/CRC Press.

Millar, Russell B. (2009). Comparison of Hierarchical Bayesian Models for Overdispersed Count Data Using DIC and Bayes’ Factors. Biometrics, 65, 962-69.

Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), March 20–22, Vienna, Austria. ISSN 1609-395X.

R Core Team (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Rubin,D. B. (1981) The Bayesian Bootstrap. The Annals of Statistics 9, 130-134. http://www.jstor.org/stable/2240875

Spiegelhalter, D.J., Best, N.G., Carlin, B.P. and Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64, 583-639. doi:10.1111/1467-9868.00353

Thouless, C.R., H.T. Dublin, J.J. Blanc, D.P. Skinner, T.E. Daniel, R.D. Taylor, F. Maisels, H. L. Frederick and P. Bouché (2016). African Elephant Status Report 2016: an update from the African Elephant Database. Occasional Paper Series of the IUCN Species Survival Commission, No. 60 IUCN / SSC Africa Elephant Specialist Group. IUCN, Gland, Switzerland. vi + 309pp

Zuur, A. F. (2019). Statistical analysis of spatial-temporal elephant poaching data using R-INLA. Prepared for CITES.